Curved spacetime bulging in time direction

[pahenning]

Let us start with a simple two-dimensional space that is asymptotically flat, like e.g. sheet of paper on a desk. Geodesics in this space are straight lines in the traditional sense.

Now add a bulge in the center of this space. without affecting the asymptotic flatness. Say, we simply distort the center of this paper sheet in the form of a half sphere. Clearly, this "space" then has a region with positive curvature - and geodesics are strangely affected. If, for example, point A is opposite from point B directly at the base of our bulge, an infinite number of equivalent geodesics exists from A to B, running as a grand (half-)circle over or around the bulge.

Now let us become somewhat weird: As above, we distort space into a locally orthogonal direction. But instead of pulling the sheet of paper into a spatial direction, we do so into the time direction. If we had an outside observer (which seems possible since we can embed our asymptotically flat spacetime into a larger flat space) he would be able to state that the center of our distorted sheet of space exists at a later time than the rest of the space. What are the geodesics in this weird space?

Moreover, some initial calculations indicate, that the "distance" between opposite points A and B at the base of our "temporal bulge" depends on the chosen way of travel. By distance of course we mean that one has to look at the proper time passing when moving from A to B at, or at least slightly under, the speed of light. In particular, the distance across the temporal bulge seems to be larger than the distance around.

Of course one has to be very careful in playing with this toy model, since in 1988 Ed Witten has shown that in (2+1)-dimensional general relativity gravitation is trivial. Nevertheless, I have become interested in this toy model and would appreciate comments.

pah

 

[Ibix]

I think you'd need to specify some maths before there was anything meaningful here. You can have curved spacelike planes in flat spacetime if you like, which seems to fit what you are describing just as well as a more complicated spacetime. You seem to be describing the geometry of your spacelike planes, but I don't see a description of how they fit together to make a spacetime.

 

[Vanadium 50]

But instead of pulling the sheet of paper into a spatial direction, we do so into the time direction

What does that mean?

 

[Orodruin]

Now add a bulge in the center of this space. without affecting the asymptotic flatness. Say, we simply distort the center of this paper sheet in the form of a half sphere.

Careful. That no longer sounds like a smooth manifold.

Clearly, this "space" then has a region with positive curvature - and geodesics are strangely affected. If, for example, point A is opposite from point B directly at the base of our bulge, an infinite number of equivalent geodesics exists from A to B, running as a grand (half-)circle over or around the bulge.

Now let us become somewhat weird: As above, we distort space into a locally orthogonal direction. But instead of pulling the sheet of paper into a spatial direction, we do so into the time direction.

What temporal direction? You were talking about a two-dimensional Riemannian manifold. It has no time direction.

In addition, as a manifold, it had no intrinsic idea about bulging out as a half sphere in three dimensions either. It just had its intrinsic geometry.

If we had an outside observer (which seems possible since we can embed our asymptotically flat spacetime into a larger flat space) he would be able to state that the center of our distorted sheet of space exists at a later time than the rest of the space. What are the geodesics in this weird space?

An embedding space is irrelevant to the actual physics of the spacetime. Unless you are trying to make an embedding where some additional physics occur in the extra dimensions, such as Kaluza-Klein theory or similar.

It is also unclear what you would even mean by bulge in the time direction. Please write down the metric you had in mind.

Moreover, some initial calculations indicate, that the "distance" between opposite points A and B at the base of our "temporal bulge" depends on the chosen way of travel. By distance of course we mean that one has to look at the proper time passing when moving from A to B at, or at least slightly under, the speed of light.

No, that is not generally what is meant by distance in GR. The typical definition would be the minimal length along the spacelike surfaces of some simultaneity foliation of the spacetime.

Lightlike geodesics are always null by definition.

 

[PeroK]

Isn't tumesence the correct term for a bulge in your manifold?

 

[Dale]

Now let us become somewhat weird: As above, we distort space into a locally orthogonal direction. But instead of pulling the sheet of paper into a spatial direction, we do so into the time direction. If we had an outside observer (which seems possible since we can embed our asymptotically flat spacetime into a larger flat space) he would be able to state that the center of our distorted sheet of space exists at a later time than the rest of the space. What are the geodesics in this weird space?

I don't think that this spacetime would satisfy the EFE.

Is there any reason not to use a standard spacetime like the Schwarzschild spacetime? If you want to use another spacetime I think that we will need an explicit expression for the metric.

 

[martinbn]

I don't think that this spacetime would satisfy the EFE.

Every spacetimes satisfies the EFE. You just pick a stress-energy tensor to match the Einstein tensor.

 

[martinbn]

Moreover, some initial calculations indicate, that the "distance" between opposite points A and B at the base of our "temporal bulge" depends on the chosen way of travel.

You don't need curvature for that, and there isn't anything surprising, it is call the twin paradox.

 

[Dale]

Every spacetimes satisfies the EFE. You just pick a stress-energy tensor to match the Einstein tensor.

Yes, good point. I think this stress energy tensor will probably violate one or more of the usual energy conditions

 

[Nugatory]

@pahenning do not overlook:

Please write down the metric you had in mind.

It is a necessary first step in reasoning about any geometry (and the only times you’ll see someone skipping it here is when the metric is already so well known that it can be referenced by name, as in “Schwarzschild” or “ADM” or the like).

If you’re not clear about what specifying the metric means and why it is so important, Google for “The parable of the surveyor” and read https://preposterousuniverse.com/wp-content/uploads/2015/08/grtinypdf.pdf

 

[Orodruin]

I don't think that this spacetime would satisfy the EFE.

Any spacetime satisfies the EFEs … with the appropriate stress energy tensor. The relevant question is if anything exists for which the stress energy tensor would take that form and whether or not certain energy conditions would be satisfied.

Every spacetimes satisfies the EFE. You just pick a stress-energy tensor to match the Einstein tensor.

Yes, good point. I think this stress energy tensor will probably violate one or more of the usual energy conditions

Doh! I need to start reading the full thread before responding …

 

[martinbn]

@pahenning do not overlook:
It is a necessary first step in reasoning about any geometry (and the only times you’ll see someone skipping it here is when the metric is already so well known that it can be referenced by name, as in “Schwarzschild” or “ADM” or the like).

If you’re not clear about what specifying the metric means and why it is so important, Google for “The parable of the surveyor” and read https://preposterousuniverse.com/wp-content/uploads/2015/08/grtinypdf.pdf

I disagree with this. For many questions you don't need to have a specific metric. For example the singularity theorems or the positive mass theorem and such. Also the explicit metrics are too few in some sense to make any general conclusion based on them.

 

[PeterDonis]

What are the geodesics in this weird space?

The only way to find out is to do the math. Write down the metric and compute its geodesics. Have you done that?

some initial calculations indicate

Please show your work.

I have become interested in this toy model and would appreciate comments.

Comments would be pointless unless you can show us the actual math.

For many questions you don't need to have a specific metric.

Even if this is the case for some questions, it is most certainly not the case for this one. Without a metric we have no idea what the OP is actually talking about, or whether the claimed spacetime even exists.

 

[pahenning]

Thank you for the comments, all are helpful, but some of them are at least a few steps ahead of me.

First of all, the example with the "bulge" in an otherwise flat sheet of paper can be considered a purely geometrical problem without talking about spacetime at all. This one is already interesting in itself, because the "bulge" can be of positive curvature (say, a half sphere) as well as of a (locally) hyperbolic nature and therefore involving a negative curvature. And considering the question of existence: Such a two dimensional manifold can be constructed from paper. I guess I'll do this just for fun.

Secondly, we hopefully can achieve consensus on the question whether in "our" universe a time direction exists. Albeit it is totally open how this could happen, and what physical effect it could eventually cause: Why should it not be possible to have an otherwise flat surface that has a deformation in this time direction?

All other questions, i.e. whether one can construct a stress-energy-tensor for this manifold, are secondary for the moment. Why do I say this? Well, consider wormholes in spacetime - they seem to be possible from purely geometrical reasons, but impossible for several physical reasons. You all know the result that a negative energy density would be needed to keep a wormhole open (a very well written paper on this is Daisuke & Hayward, Phys.Lett. A260 (1999) 175-181, https://doi.org/10.48550/arXiv.gr-qc/9905033).

pah

 

[PeterDonis]

the example with the "bulge" in an otherwise flat sheet of paper can be considered a purely geometrical problem without talking about spacetime at all

Sure, but then it belongs in the math forum, not the relativity forum. So if that's what you want to talk about, then you need to open a new thread in the math forum and this one will be closed.

we hopefully can achieve consensus on the question whether in "our" universe a time direction exists.

There are certainly timelike curves and timelike vectors in our universe, yes. But that is not what your OP is talking about.

Why should it not be possible to have an otherwise flat surface that has a deformation in this time direction?

Nobody is saying it is not possible. We are asking you to write down the actual metric for such a thing, or else give a reference. Without a metric there is nothing we can discuss.

All other questions, i.e. whether one can construct a stress-energy-tensor for this manifold, are secondary for the moment.

They certainly are, but not for the reason you give--for the simple reason that until we have an actual metric, we cannot even talk about your question at all, because we have nothing on which to base a discussion.

consider wormholes in spacetime - they seem to be possible from purely geometrical reasons, but impossible for several physical reasons.

Yes, but the only reason we can even talk about such things for wormholes is that we have actual wormhole metrics published in the literature on which to base the discussion. That's how we know, for example, that exotic matter (a better term than your "negative energy density") is necessary to hold a wormhole open: we can compute that from the published wormhole metrics.

Do you have an actual metric for your proposal? If you don't, we have no basis for discussion and this thread will be closed.

 

[Orodruin]

Why should it not be possible to have an otherwise flat surface that has a deformation in this time direction?

This cannot be answered until you define what you mean by a ”deformation in the time direction”. Curvature is described by the Riemann curvature tensor, whose (anti)symmetries require two directions for its components to be non-zero. A single direction is not sufficient for curvature to exist.

 

[Ibix]

I think what @pahenning is envisaging is something like this:

We start with a flat 2d sheet embedded in 3d Euclidean space. Then we displace some set of points in the direction perpendicular to the sheet. A section through the result would look like

Then we change to a flat spacelike sheet embedded in (2+1)d Minkowski spacetime and do the same - the same diagram as above could be drawn, except with $y$ replaced by $t$. Then we clone those sheets and stack them to form a spacetime:

Then we can ask questions about this spacetime.

Is that right, @pahenning?

If so, there are a number of issues that I see. First, you can't insert a hemisphere this way because the edge of the hemisphere is orthogonal to the plane and, when we get to stacking, it will intersect the plane above. Also, the plane would be timelike in part of it and null in at least one circle. There are limits to the displacements you can apply (in any Lorentzian signature spacetime, not just Minkowski) if you want your planes to remain spacelike. Second, the metric on the plane is induced from the metric of the space in which it is embedded. Is that meant to be flat Minkowski spacetime, or something else? Third, when we stack the planes, what is the "distance" (what ADM would call lapse and shift) between them?

All of this affects the spacetime you get. For example, if your embedding space is flat Minkowski spacetime and you specify (ref. my second diagram) vertical shift vectors and a constant lapse you just get flat Minkowski spacetime in weird coordinates. Other choices will give you other spacetimes. So there's quite a lot to answer before we can actually do anything with your model.

 

[A.T.]

Now let us become somewhat weird: As above, we distort space into a locally orthogonal direction. But instead of pulling the sheet of paper into a spatial direction, we do so into the time direction.

Yeah, but that is hard to visualize. Usually you have to drop all but one spatial dimension to show the distorted time dimension with an embedded manifold. Then you indeed get a "temporal bulge". See:

Chapter 10 here:
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/page/n165/mode/2up

Chapter 2.5 here:
https://www.relativitet.se/Webtheses/tes.pdf

 

[Hill]

I think that the metric in the spherical part of the bulge, instead of the usual $d \theta ^2 + sin^2 \theta \, d \phi ^2$, becomes $cos 2 \theta \, d \theta ^2 + sin^2 \theta \, d \phi ^2$.

 

[Orodruin]

I think we should all stop speculating in what the OP means until they have specified it themselves. Speculation is meaningless and possibly misleading until we have an actual clarified question.

 

[Bosko]

Every spacetimes satisfies the EFE. You just pick a stress-energy tensor to match the Einstein tensor.

In that case, the gravitational flux would not be conserved.

 

[Bosko]

Now add a bulge in the center of this space. without affecting the asymptotic flatness. Say, we simply distort the center of this paper sheet in the form of a half sphere.

The edge of a half sphere is not smooth. The spacetime should be smooth (Reimannian manifold)

But instead of pulling the sheet of paper into a spatial direction, we do so into the time direction.

Does the speed of light remain the same then?

 

[Dale]

And considering the question of existence: Such a two dimensional manifold can be constructed from paper.

Such a manifold would be Riemannian, not pseudo Riemannian. Meaning there would be no time dimension.

But even with such a manifold I don’t know how to determine what the geodesics look like without a metric. Maybe someone else with more experience in Riemannian geometry would know that intuitively. Personally, I am not at that level, I would need a metric to answer any of the questions you asked, as well as my own concerns.

Good luck, I will have to bow out at this point.

 

[pahenning]

Concerning the metric: I can let you know when I have it published (which I intend to do). Otherwise, I thank all contributors again, I have indeed profited from some of the constructive comments (Yes, Ibix, this is what I mean - but not static stacking). But with all due respect I won't waste my lifetime on discussing whether my question has been appropriate for the forum.

 

[martinbn]

Concerning the metric: I can let you know when I have it published (which I intend to do). Otherwise, I thank all contributors again, I have indeed profited from some of the constructive comments (Yes, Ibix, this is what I mean - but not static stacking). But with all due respect I won't waste my lifetime on discussing whether my question has been appropriate for the forum.

It is not about appropriatness, it is about clearity. No one here has any idea what you are asking. Everyone is just guessing because of they way you asked the question.

 

[renormalize]

In that case, the gravitational flux would not be conserved.

I don't follow. Can you expand on this?

 

[Orodruin]

Concerning the metric: I can let you know when I have it published (which I intend to do). Otherwise, I thank all contributors again, I have indeed profited from some of the constructive comments (Yes, Ibix, this is what I mean - but not static stacking). But with all due respect I won't waste my lifetime on discussing whether my question has been appropriate for the forum.

If you try to publish something as vague and unspecific as what you have presented here … it won’t get published. At least not in any notable journal.

 

[Orodruin]

spacetime:

Yes, Ibix, this is what I mean

That’s just a non-standard foliation of Minkowski space. A different set of curvilinear coordinates in special relativity with seemingly no particular usefulness.

 

[pahenning]

Just wait for the publication before denigrating it...

 

[Orodruin]

Just wait for the publication before denigrating it...

I am not denigrating anything. I am just saying that what Ibix presented was a foliation of regular Minkowski space. Either that is a description of what you are trying to do or it isn’t. If it is, then it is uninteresting. If it isn’t, then you should not say that it is. We have no way of telling which since you refuse to give any actual details that could actually elucidate what you want to know.

 

[PeterDonis]

Concerning the metric: I can let you know when I have it published (which I intend to do).

Just wait for the publication before denigrating it...

In other words, you don't currently have a published metric describing what you're talking about. In that case we have no valid basis for discussion.

Thread closed.